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CONTENTS

Introduction to View Transformation

Viewing Transformation is the mapping of coordinates of points and lines that form the picture into appropriate coordinates on the display device.

  World coordinate system (WCS) is the right handed cartesian co-ordinate system where we define the picture to be displayed.   A finite region in the WCS is called the Window. The corresponding coordinate system on the display device where the image of the picture is displayed is called the physical coordinate system. Mapping the window onto a subregion of the display device called the viewport is called the Viewing Transformation. Normalized device coordinate (NDC) is the display area of the virtual display device corresponding to a unit square. The lower left handed corner is the origin of the coordinate system. Mapping the window in the world coordinate space to viewport in NDC space is called the Normalization Transformation, N.

where is the scale in x and y directions, given by


Introduction to View Transformation View Coordinate System

View Transformation

A 3D scene can be viewed from any posistion in 3D space. A "synthetic" camera positioned and oriented in 3D space can be used to describe the viewing, and the part of the image or scene to be viewed . It has the following three principal ingredients.

  1. View plane: The window is defined in this plane.
  2. View coordinate system: Usually a left handed system called the UVN system is used.
  3. An eye defined within this system. and e determines a plane orthogonal to containing e.

Let be the look-at-point. For perspective views, the view plane normal as a unit vector from eye to a ``look-at point'' is given by

The view up vector is the tilt (rotation) of the head or camera.

For parallel views it is convenient to think of the view plane normal as determining the direction of projection.


View Coordinate System

An object in world coordinate space, whose vertices are can be expressed in term of view coordinates .

The viewing transformation is


Example of View Transformation


Alternative Construction of View Transformation

Proof

Its along z axis. Hence it can be concluded that the mathematical computation is right.


Excercise

  1. Find the viewing transformation matrix given a view reference point of , a look-at point , and a view-up vector .
  2. Find the viewing transformation given eye position , look-at point and up-vector .
  3. Find the viewing transformation given eye position , look-at point and up-vector .
  4. Find the viewing transformation given eye position , look-at point and up-vector .
  5. If and are two different rows of a rotation matrix (a) what is their inner product, (b) what can be said about their cross product?

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Priya Asokarathinam /ADVISOR Shoaff